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Staff Working Paper/Document de travail du personnel 2016-42
On the Value of Virtual Currencies
by Wilko Bolt and Maarten R.C. van Oordt
2
Bank of Canada Staff Working Paper 2016-42
August 2016
On the Value of Virtual Currencies
by
Wilko Bolt1 and Maarten R.C. van Oordt2
1Economics and Research Division De Nederlandsche Bank
Amsterdam, The Netherlands [email protected]
2Financial Stability Department
Bank of Canada Ottawa, Ontario, Canada K1A 0G9
ISSN 1701-9397 © 2016 Bank of Canada
i
Acknowledgements
We are grateful to Ben Fung, Rodney Garratt, Hanna Halaburda, Kim Huynh, Charles Kahn, Andrew Levin, Paul Metzemakers, Radoslav Raykov, Rune Stenbacka, Marianne Verdier, Sweder van Wijnbergen, as well as participants of the Annual Conference of the Bank of Canada on “Electronic Money and Payments” (Ottawa, 2015), the 14th Annual International Industrial Organization Conference (Philadelphia, 2016), De Nederlandsche Bank Payments Conference on “Retail Payments: Mapping Out The Road Ahead” (Amsterdam, 2016), the 91st Annual Conference of the Western Economic Association International (Portland, Oregon, 2016), the World Finance Conference (New York, 2016), and research seminars at the Bank of Canada (2015) and De Nederlandsche Bank (2016) for useful comments and suggestions. This research project was carried out partly while the second author was employed by De Nederlandsche Bank. Views expressed are those of the authors and do not necessarily reflect those of the European System of Central Banks, De Nederlandsche Bank or the Bank of Canada.
ii
Abstract
This paper develops an economic framework to analyze the exchange rate of virtual currency. Three components are important: first, the current use of virtual currency to make payments; second, the decision of forward-looking investors to buy virtual currency (thereby effectively regulating its supply); and third, the elements that jointly drive future consumer adoption and merchant acceptance of virtual currency. The model predicts that, as virtual currency becomes more established, the exchange rate will become less sensitive to the impact of shocks to speculators’ beliefs. This undermines the notion that excessive exchange rate volatility will prohibit widespread use of virtual currency.
Bank topics: Asset pricing; E-money; Exchange rates JEL codes: E42, E51, F31, G1
Résumé
Nous construisons un modèle d’analyse pour cerner les facteurs qui influent sur le taux de change des monnaies virtuelles. Trois éléments importent : tout d’abord, l’utilisation actuelle d’une monnaie virtuelle comme moyen de paiement; ensuite, la décision d’investisseurs prospectifs d’acheter de la monnaie virtuelle (et de contrôler ainsi l’offre de ce type de monnaie); enfin, les facteurs qui détermineront ensemble l’adoption d’une monnaie virtuelle par les consommateurs et son acceptation par les commerçants. Selon les prévisions du modèle, lorsqu’une monnaie virtuelle devient plus établie, son taux de change devient moins sensible aux répercussions des changements non anticipés de croyance des spéculateurs. Ce résultat va à l’encontre de l’idée voulant que la très forte volatilité de leur taux de change empêchera la large diffusion des monnaies virtuelles.
Sujets : Évaluation des actifs; Monnaie électronique; Taux de change Codes JEL : E42, E51, F31, G1
Non-Technical Summary
This paper develops a theoretical framework to analyze the economic factors affecting the
exchange rate for virtual currency. Virtual currencies, such as Bitcoin, represent both the
emergence of a new form of currency and a new payment technology to purchase goods and
services. These currencies may move outside the scope of current financial institutions, as
they allow distant payments to be made directly between consumers and merchants without
the use of any financial intermediaries. Moreover, their supply is not necessarily controlled
by central banks, and speculative motives are widely believed to be an important factor for
the value of virtual currencies.
Speculating on the value of currencies is not new. In the early 1900s, Fisher (1911)
argued that, in certain situations, speculators may effectively regulate the money supply by
withdrawing money from circulation by betting on its future value. We apply this “old”
notion in a formal way by using Fisher’s quantity relation and showing how the exchange
rate of a virtual currency responds to changes in the speculative position of investors.
Our theoretical framework shows that three components are important for the exchange
rate. First, the current use of virtual currency to make actual payments. Second, the de-
cision of forward-looking investors to buy virtual currency (thereby effectively regulating
its supply). Third, the elements that jointly drive future consumer adoption and merchant
acceptance of virtual currency. On the consumer side of the market, private benefits may be
large for those who frequently execute cross-border payments, such as remittances. Addition-
ally, consumers who value privacy and anonymity more, and those who are technologically
more adept are likely to gain from using virtual currencies. On the other side of the market,
large merchants may experience considerable private benefits from avoiding the high fees
charged by traditional payment providers. Internet stores may gain as well, since they face
relatively low implementation costs when accepting virtual currencies.
The model predicts that, as virtual currency becomes more established, the exchange
rate will become less sensitive to the impact of shocks to speculators’ beliefs and their inflow
into and outflow from the virtual currency market. This prediction undermines the notion
that the current high volatility of the exchange rates of virtual currencies will prohibit their
widespread usage in the long run.
“You have to really stretch your imagination to infer what theintrinsic value of Bitcoin is. I haven’t been able to do it.
Maybe somebody else can.”– Alan Greenspan, Bloomberg Interview, 4 December 2013
1 Introduction
This quote from the former Federal Reserve Chairman could hardly be more accurate in
describing the aim of the present study, in which we attempt to answer the broad question
of what drives the value of virtual currencies. Virtual currencies, such as Bitcoin, represent
both the emergence of a new form of currency and a new payment technology to purchase
goods and services. These currencies may move outside the scope of current financial insti-
tutions. That is, their supply is not necessarily controlled by central banks, and they allow
distant payments to be made directly between consumers and merchants without the use
of any financial intermediaries. The key innovation is the implementation of cryptographic
identification techniques into a “distributed ledger,” i.e., a digital record that allows the
tracking and validation of all payments made. This allows virtual currencies to be used in a
decentralized payment system while avoiding the possibility of “double spending.”
Bitcoin is the most well-known virtual currency.1 For primers on the economics behind
Bitcoin, see, e.g., Dwyer (2015) and Bohme et al. (2015). Bitcoin was launched in 2009 and
attracted attention from the financial press, economists, central banks and governments.
This attention was fuelled by the sudden “explosion” and volatility in the exchange rate of
Bitcoin by the end of 2013. In November 2013, the US-dollar exchange rate for one unit
of Bitcoin increased more than fivefold. Bitcoins had begun trading for less than a nickel
in 2010; by November 2013, the exchange rate exceeded $1,100. During 2014, however, the
exchange rate quickly lost ground again, settling at around $250 in March 2015, after which
its value rose to $650 per bitcoin in June 2016. While the supply of Bitcoin units over time is
mathematically specified with an upper limit of 21 million units, its current supply amounts
1See, e.g., Ong et al. (2015) and Tarasiewicz and Newman (2015) for a description of alternative virtualcurrencies and their designs.
1
to approximately 15.7 million units (in June 2016). Bitcoin’s usage is limited but rising:
from around 20,000 daily transactions in 2012 to over 70,000 daily transactions in 2014, and
reaching over 200,000 daily transactions on average in June 2016. All in all, compared with
the volume and value of other existing currencies, Bitcoin is still a relatively small monetary
phenomenon, but it has been growing.
This paper develops an economic framework that analyzes the exchange rate of virtual
currency at its early-adoption stage and its main drivers. A unique combination of the
properties of virtual currencies – at least in this early stage – plays an important role in
our model. First, virtual currency prices of products and services are perfectly flexible
with respect to changes in the exchange rate, since merchants tend to instantly adjust price
quotes in virtual currency to the latest available exchange rate. In the model, this property
is key in providing a direct link between the exchange rate and the demand for virtual
currency. Second, the choice for making payments with virtual currency is also a choice for
an alternative transaction technology, since these payments are settled and processed through
a peer-to-peer payment network associated with that virtual currency. Network economies
affecting payment choice play an important role in determining the ultimate demand for
virtual currency. Third, the growth of the supply of virtual currency is, to a large extent,
predetermined. In line with this latter characteristic, future demand for virtual currency to
execute payments is one of the main sources of uncertainty in our model.
Perfect flexibility of prices with respect to the exchange rate is not a unique feature of
virtual currency in the early-adoption stage. It may also characterize a traditional currency
facing extremely high and volatile inflation, when price setting occurs in practice in terms
of a stable (foreign) currency (such as the US dollar); see, e.g., Dornbusch et al. (1990).
However, an important difference is that investors have no incentives to hold a highly in-
flationary currency as a store of value, while speculative motives are widely believed to be
one of the reasons for holding virtual currency. Hence, from a monetary perspective, virtual
currency in the early-adoption stage provides an interesting case study for simultaneously
2
observing speculative demand for a currency and perfect flexibility of prices with respect to
the exchange rate.
Our framework combines an investor’s portfolio model with a payment network model,
while adding a flavour of monetary economics. In our framework, three components are
important for the exchange rate: (i), the actual use of virtual currency to execute real
payments; (ii), the decision of forward-looking investors to buy virtual currency (thereby
effectively regulating its supply); and (iii), the elements that jointly drive future consumer
adoption and merchant acceptance of virtual currency. These latter elements determine
the expected long-term growth in virtual currency usage. We show that the equilibrium
exchange rate is determined by both a “purely speculative” component that depends on the
hypothetical price speculators would offer if not a single real transaction is settled using
virtual currency, and a transaction component that depends on the actual amount necessary
to facilitate real payments.
Speculating on the value of currencies is not new. In the early 1900s, Fisher (1911)
argued that, in certain situations, speculators may effectively regulate the money supply by
withdrawing money from circulation by betting on its future value. We apply this “old”
notion in a formal way by showing that the exchange rate of a virtual currency immediately
responds to changes in the speculative position of investors. Our model predicts that, as a
virtual currency becomes more established, the exchange rate will become less sensitive to
the impact of shocks to speculators’ beliefs and their inflow into and outflow from the virtual
currency market. This prediction undermines the notion that the current high volatility of
the exchange rates of virtual currencies will prohibit their widespread usage in the long run.
Additionally, we borrow from the so-called “two-sided markets” literature to explain the
main factors that drive future consumer adoption and merchant acceptance of virtual cur-
rency as a payment instrument; see, e.g., Armstrong (2006) and Rochet and Tirole (2006). It
is shown that private benefits and cross-group externalities among merchants and consumers
affect the joint demand for virtual currency to make payments for real goods and services. On
3
Figure 1: Speculation and the exchange rate of Bitcoin, USD
2011 2012 2013 2014 2015
020
040
060
080
010
00
The figure presents the value of a unit of the virtual currency Bitcoin. The solid black line shows theexchange rate. The grey line shows a rough estimate of the hypothetical exchange rate if no units of thevirtual currency were held for speculation. The dashed line provides an impression of the potential exchangerate in the absence of speculation and is a strongly smoothed average of the grey line. For technical details,see Appendix A. Source: www.blockchain.info and authors’ calculations.
one side of the market, private benefits may be large for consumers who frequently execute
cross-border payments, such as remittances, that usually carry high fees. Face-to-face pay-
ments may also become easier and cheaper via application-based “wallets” on a smartphone.
Consumers who value privacy and anonymity more, and those who are technologically more
adept are likely to gain from using virtual currencies. On the other side of the market, large
merchants may experience considerable private benefits from avoiding the high fees charged
by traditional payment providers. Internet stores may gain as well, since they face relatively
low implementation costs when accepting virtual currencies. Our model tries to explain how
the resulting joint demand for virtual currency affects its exchange rate.
A first impression of what the exchange rate of Bitcoin might have looked like in the
absence of speculation is presented by the dashed line in Figure 1. The solid black line
shows daily data on the actual exchange rate. Based on our framework, the grey line shows a
downward correction of the actual exchange rate to account for the virtual currency positions
4
held by speculators. Here, the size of the speculative position is roughly proxied by the
number of bitcoins that will remain in “dormant” accounts for an extended period. Economic
theory, such as buffer stock models, suggest that consumers and merchants do not instantly
adjust their positions to their actual liquidity needs in response to fluctuations; see, e.g.,
Laidler (1984) and Mizen (1997). This suggests that only a strongly smoothed version of
the grey line, such as the dashed line, may provide some impression of the exchange rate in
the absence of speculation.
The paper is set up as follows. In Section 2, we briefly review the literature. As a
preliminary, Section 3 analyzes the exchange rate of virtual currency using Fisher’s quantity
equation. Section 4 describes the model, building on a two-sided payment network model
and an investor’s portfolio model. In Section 5, the rational expectation equilibrium results
are presented and explained. Various extensions are discussed in Section 6. Finally, Section
7 provides some discussion and concluding remarks.
2 Literature
While most papers focus on the technical and computational aspects of creating virtual
currencies, the literature on the economics of virtual currencies is still developing.
In a recent paper, Dwyer (2015) provides an overview of private virtual currencies with
a special focus on Bitcoin (being the most prominent example). The behaviour of Bitcoin’s
price since it began trading on electronic exchanges is analyzed, and the author provides
a comparison of the volatility of Bitcoin’s price on those exchanges compared with gold
and foreign exchange. He argues that a theoretical model pinning down a dynamic reputa-
tional equilibrium for the central mechanism of Bitcoin’s functioning is needed for a deeper
fundamental understanding of Bitcoin and similar currencies.
In another study, Bohme et al. (2015) present Bitcoin’s design principles and properties,
and review its past, present, and future uses. The authors point out risks and regulatory
5
issues as Bitcoin interacts with the conventional financial system and the real economy.
The authors acknowledge the deflationary risk of the fixed money-growth rule inherent in
Bitcoin’s design, and point to the difficulty as to whether decentralized virtual currencies
can be designed with monetary policies that include feedback or discretion.
The rapid appreciation of the exchange rate and its high volatility have been posed as
major concerns for the viability of Bitcoin’s use as a currency. Yermack (2015) examines
its historical trading behaviour to analyze whether it behaves like a traditional existing cur-
rency. Bitcoin suffers from much higher exchange rate volatility than the exchange rates of
traditional currencies. Moreover, based on a purely statistical analysis of the Bitcoin ex-
change rate, Cheung et al. (2015) and Cheah and Fry (2015) express their concern regarding
its “bubble-like” behaviour; see also Weber (2016). Moreover, Yermack (2015) documents
Bitcoin’s exchange rate to show virtually zero correlation with other existing currencies, re-
ducing Bitcoin’s use for risk-management purposes and making it exceedingly difficult for
its owners to hedge. Bitcoin also lacks access to a banking system with deposit insurance,
and it is not used to denominate consumer credit or loan contracts. In this context, Bitcoin
appears to behave more like a highly speculative investment than a currency.
Schuh and Shy (2016) report that, according to the 2014-2015 Survey of Consumer Pay-
ment Choice (SCPC), about half of US consumers are aware of Bitcoin. About one percent
or less of US consumers have ever owned (adopted) virtual currency, and most of them used
it recently to pay a person (most common) or a merchant. Expectations of future price seems
to be the biggest driver of adoption, and those consumers who are especially “interested in
new technologies” are more likely to adopt. See Badev and Chen (2014) for more on the
technical background and empirical regularities of Bitcoin.
Ali et al. (2014) examine the economic incentives of adopting virtual currencies and assess
potential risks to monetary and financial stability. A key attraction of virtual currencies at
present is their low transaction fees. But these fees may increase as usage grows and may
eventually be higher than those charged by incumbent payment systems. They also discuss
6
various factors that influence the value of virtual currencies, such as risk-return trade-offs,
transaction costs and relative benefits, and habit formation.2
In an excellent primer on the mechanics of Bitcoin, Velde (2013) regards Bitcoin as
an elegant implementation of virtual currency – controlling its creation and avoiding its
duplication simultaneously – but questions whether it can truly rival or replace existing
currencies. The paper argues that, should Bitcoin become widely accepted, it is unlikely
that it will remain free of intervention by public authorities, if only because the governance
of the Bitcoin computational code and “mining” protocol is opaque and vulnerable.3
Network effects may play an important role in the adoption of a virtual currency.
Ha laburda and Sarvary (2015) describe the emergence of virtual currency in the histori-
cal context of an ever-present competition among different forms of money. In the context of
competition among virtual currencies, Gandal and Ha laburda (2014) empirically investigate
the presence of a “winner-takes-all” effect for the most popular virtual currency. Their data
suggest that this effect is only dominant early in the market, but this trend is reversed in a
later period, which may be consistent with the use of virtual currencies as financial assets.
Moreover, Sauer (2015) also discusses the demand for virtual currency in the context of
(one-sided) network effects.
Most papers lack a formal treatment of the economics behind the exchange rate of virtual
currencies, unveiling the links between speculative behaviour, currency creation, network
effects, and real growth in transactions for goods and services. This paper tries to bridge
that gap.
2Similarly, in an earlier report, the European Central Bank (2012) assessed the potential economic impactof virtual currency schemes for central banks, covering price stability, financial stability and risk to paymentsystems. The risks were deemed low because of their limited connection with the real economy, their lowvolume traded and current lack of wide user acceptance. Other publications in outlets of central bankscommenting on the potential risks and use of virtual currencies include, e.g., Arias and Shin (2013), Beerand Weber (2014), Segendorf (2014), Tasca (2015) and Young (2015).
3In a more technical paper, Kroll et al. (2013) model a game between Bitcoin miners and Bitcoin holdersand characterize a multiplicity of equilibria. In particular, they show how, in equilibrium, a “Goldfinger-style” attacker might be able to disrupt the Bitcoin system, “crashing” the currency and destroying its value.As in the famous James Bond (1964) movie, the villain Auric Goldfinger wants to increase the value of hisown gold holdings by making the gold in Fort Knox radioactive and worthless, thereby undermining the(gold-backed) US currency.
7
3 Model Preliminaries: Quantity Equation
In the model in Section 4, we consider two currencies: an established currency, say e ,
and a virtual currency, say B. But before we proceed, we first turn to the well-known
transactions version of the quantity equation popularized by Fisher (1911), i.e.,
PBt TB
t = MBt V B
t , (3.1)
where V Bt denotes the velocity of virtual currency B, defined as the average number of times
each unit of virtual currency is used to purchase real goods and services within period t. TBt
is the quantity of real goods and services purchased with virtual currency B during period t,
and PBt is the weighted average price. MB
t denotes the (nominal) quantity of money defined
as the number of units of virtual currency B. Eq. (3.1) holds by definition within any given
period t; see, e.g., Friedman (1970). Essentially, the expression follows directly from defining
the concept of the velocity of virtual currency B. It does not depend on any behavioural
assumptions.
Some manipulation of the left-hand side of Eq. (3.1) gives
PBt
P et(P et TB
t ) = MBt V B
t , or,PBt
P etTB∗t = MB
t V Bt , (3.2)
where P et denotes the weighted average price of the goods and services purchased with virtual
currency when quoted in the established currency, and where TB∗t denotes the volume of trade
in goods and services with payments settled in virtual currency B, where the asterisk in the
superscript signifies that this quantity is now measured in terms of the established currency.4
4Note that it is a simplification to interpret the index of the general price level as the empirical counterpartof Pet . It is easily verified that this would hold if the basket of goods used to calculate the index of thegeneral price level is the same as the basket of goods for which payments using virtual currency are made.The differences between the baskets may be more pronounced if virtual currency is used for certain nicheproducts and services, such as the market for electronics.
8
Although (3.2) is, for simplicity, derived for a single established currency, it can easily be
extended to cover multiple established currencies instead.
In the model, we essentially assume the established currency to be the unit of account.5
Let Se /Bt denote the exchange rate, i.e., the number of units of the established currency one
pays to obtain a single unit of virtual currency in period t. We assume that the prices of
goods and services expressed in virtual currency are completely determined by the exchange
rate and their price level in the traditional currency as PBt = P et /S
e /Bt . This interpretation
connects well with the practice of many online stores to adjust prices quoted in virtual
currencies instantly to the latest available exchange rate. The prices in these stores expressed
in virtual currency are perfectly flexible with respect to changes in the exchange rate. Using
this assumption in Eq. (3.2) yields the exchange rate for any t as
Se /Bt =
TB∗t
MBt V B
t
. (3.3)
Moreover, without loss of generality, the average velocity, the V Bt , can be rewritten as
a weighted average of the velocity of the units of virtual currency used to settle payments
for goods and services, V B∗t , and the velocity of those not used to settle the payments for
goods and services. Note that the latter equals zero, since velocity was defined as the average
number of times each unit of virtual currency is used to purchase real goods and services.
Formally,
V Bt =
MBt − ZB
t
MBt
V B∗t +
ZBt
MBt
0, (3.4)
where ZBt ≥ 0 is the number of units of virtual currency not used to settle the payments for
goods and services during period t. Essentially, ZBt units are “stored value” and we sugges-
tively refer to ZBt as the “speculative position” in virtual currency. After all, speculators buy
units of virtual currency in the hope of making a profit by selling them in the future. Such a
strategy involves only the exchange of the established currency against the virtual currency
5This is an unrealistic assumption for an economy where virtual currency would take the role of theestablished currency.
9
and does not involve the use of virtual coins for the payment of real goods or services. These
speculators may include both “pure” speculators, who do not use virtual currency for any
real transaction, as well as merchants and consumers who hold larger positions in virtual
currency than is demanded to execute payment transactions for goods and services.
Combining Eqs. (3.3) and (3.4) gives the level of the exchange rate as
Se /Bt =
TB∗t
(MBt − ZB
t )V B∗t
. (3.5)
This equation describes the effect of three factors affecting the exchange rate of virtual
currency that are common in the context of the value of money: the exchange rate increases
in the volume of the payments for goods and services with virtual currency, TB∗t ; and it
decreases in the velocity of virtual currency, V B∗t , and in the total quantity of virtual currency,
MBt .
The expression in (3.5) also includes a fourth factor affecting the exchange rate of virtual
currency, which is the quantity of virtual currency held in the speculative position, ZBt .
Eq. (3.5) shows that the speculative position effectively reduces the quantity of virtual
currency available to facilitate real payments and, therefore, increases the value of virtual
currency. The impact of speculation on the exchange rate is shown in Figure 2. In the
absence of speculation, the exchange rate for virtual currency would be determined only
by the intersection between the curve and the vertical axis in Figure 2. This point on the
curve corresponds to the case in which all units of virtual currency are used to facilitate
payments for real goods and services. In the presence of speculation, the exchange rate is
higher because fewer units of virtual currency are available to facilitate real payments.
Of course, assigning a role to speculation in determining the value of money is an old
notion. For example, Fisher (1911) mentioned the relation between speculation and the
effective quantity of money in the context of the redemption of greenbacks by the US gov-
ernment from 1879 onwards. In the run-up to the promised redemption of greenbacks, he
10
Figure 2: Speculation and the value of virtual currency
0 4 8 12
04
812
MtBZt
B
St€
B
Speculative position Transactions
The figure shows the value of a unit of virtual currency, Se /Bt , as a function of the speculative position,
ZBt , for some fixed level of TB∗
t /V B∗t , such as described by Eq. (3.5). The size of the speculative position is
limited from above by the total number of units of virtual currency, MBt , as indicated by the dashed vertical
line. The units of virtual currency not held for speculative investment, i.e., MBt − ZB
t , are in circulation toaccommodate payments.
wrote, “[s]ome of them were withdrawn from circulation to be held for the rise. (...) Thus
speculation acted as a regulator of the quantity of money.” (p. 261). This role of speculation
is formalized in Eq. (3.5). It shows how the exchange rate of virtual currency responds
immediately to changes in the magnitude of the speculative position as a consequence of
assuming that prices quoted in virtual currency are perfectly flexible with respect to the
exchange rate. With fewer units of virtual currency available because of speculation, an
immediate rise in the exchange rate is necessary to facilitate a given volume of transactions
in goods and services in terms of the established currency.
Figure 1 in the Introduction, which provides some impression of the exchange rate of
Bitcoin in the absence of speculation, is based on the expression in (3.5). The observed
exchange rate Se /Bt is presumed to be the result of both real transactions and speculation,
where the latter is measured by a rough proxy, i.e., the units of virtual currency that will
11
remain in dormant accounts for an extended period. For each point in time, it is subsequently
possible to determine a hypothetical exchange rate in the absence of speculation by the
intersection of the curve and the vertical axis in Figure 2. The intersection provides a rough
estimate of the exchange rate if all units of virtual currency were in use to facilitate payments
for goods and services, and is presented for each point in time by the solid grey line.
4 Model
The model contains two building blocks. The first building block is based on two-sided
market theory that tries to identify which factors drive the uptake of virtual currency as a
payment instrument.6 The more consumers are willing to transact using virtual currency, the
more merchants may wish to accept this type of currency for payment. And, vice versa, the
more merchants are willing to accept virtual currency, the more likely that consumers will
start using it. These two-sided cross-group externalities play an important role for optimal
pricing and therefore the use of virtual currency to make payments.
The second building block concerns the behaviour of speculators. The investment decision
of a speculator is modelled as a trade-off between investing in a risk-free bond denominated
in the established currency or speculation on the future value of virtual currency.
4.1 Model set-up
Essentially, the model is a one-period model. Period t refers to the initial state; period
t + 1 refers to the steady state. Regarding the steady state, for simplicity, we consider the
following scenario. Given technological uncertainties, potentially adverse regulatory policies,
or successful introductions of other virtual currencies, two extreme events may occur at
t+1. With probability q, the virtual currency payment network will end up in its stationary
equilibrium. Alternatively, with probability 1 − q, the payment network will be abandoned,
6See, e.g., Armstrong (2006) and Rochet and Tirole (2006) for a deeper understanding of two-sidedmarkets.
12
in which case, the units of virtual currency will be worthless. In the case of success with
probability q, the stationary equilibrium is such that the frequencies of virtual currency users
(i.e., consumers and merchants) are equal to their equilibrium values.7
Lastly, similar to the Bitcoin supply process in practice, we assume that the number
of virtual currency units at t + 1 follows a predetermined growth rule, such that MBt+1 =
(1 + mBt+1)M
Bt .
4.2 Two-sided virtual currency payment network
Organized as an electronic payment network, the underlying idea is that virtual currency
creates economic value by enabling transactions between merchants and consumers. Assume
that consumers and merchants derive net utility from using a monopolistic virtual currency
network
Ui,t = αiNj,t + βi − pi,t, i, j = c,m, i 6= j, (4.1)
where Nj,t is the number of users from the other side who use the virtual currency network
in period t, αi > 0 is the benefit that user i enjoys from transacting with each user on
the other side, βi is the fixed benefit the user obtains from connecting to the network, and
pi,t is the fixed “membership” fee that is levied by the network as a lump-sum charge.8
The users of virtual currency are assumed to be heterogeneous in their fixed benefit for
network services. This heterogeneity is described by a cumulative density function Fi(·)
with probability density function fi(·), i = m, c.
On the consumer side, the benefits from joining a virtual currency network may be larger
for more technologically adaptive consumers. Such consumers may incur a lower cost when
7In their seminal paper, Kareken and Wallace (1981) show that the nominal exchange rate between twoperfectly substitutable “fiat” currencies is indeterminate. This indeterminacy result would not hold in ourcase since the two currencies are different in terms of their risk profiles as well as in terms of the liquidityservices provided by the virtual currency payment network.
8Although, for simplicity, it is assumed here that the utility Ui is linear in the number of users Nj ,specifying a non-linear relation would not qualitatively change the results (Armstrong, 2006; Anderson,2012). However, despite the linear specification, the number of consumers Nc generally increases with thenumber of merchants Nm in a non-linear way, and vice versa.
13
adopting new payment technologies. Benefits may also be larger for consumers who face
higher costs from traditional payment networks for a variety of reasons. Costly cross-border
payments, such as sending remittances, may be one such reason. Privacy and anonymity
may also increase the private benefits from using virtual currencies. On the merchant side,
heterogeneity in the benefits from joining a virtual currency network may depend on various
factors, such as merchant size and distribution channels. It is well documented that large
retailers pay high merchant service fees for accepting certain credit and debit cards. Re-
garding distribution and business models, online stores may face lower implementation costs
from accepting virtual currency than traditional stores.
Based on the utility-maximizing behaviour of potential users, the number of users i that
connect to the network is given by
Ni,t = N iDi(Ui,t ≥ 0) = N i Pr(βi ≥ pi,t − αiNj,t) = N i
(
1 − Fi(βi,t))
, (4.2)
where N i is the maximum number of potential users of type i, and where βi,t
= pi,t−αiNj,t.
Given costs Ci that are incurred by the network when users join the virtual currency
network, (per-period) total network profits are given by
πt(pc,t, pm,t) = Nc,t(pc,t − Cc) + Nm,t(pm,t − Cm). (4.3)
Substituting pi,t = αiNj,t + βi,t
in Eq. (4.3), for network profits, we write:
π(βc,t, β
m,t) =N cDc(βc,t
)(
αcNmDm(βm,t
)βc,t
− Cc
)
+
NmDm(βm,t
)(
αmN cDc(βc,t) + β
m,t− Cm
)
. (4.4)
Thus, network profits can be expressed as a function of only (βc,t, β
m,t).
The interior solution (β∗
c, β∗
m) that solves the first-order conditions in turn determines
profit-maximizing numbers (N∗c , N
∗m) of virtual currency users and profit-maximizing fees
14
(p∗c , p∗m). Profit-maximizing network fees satisfy
p∗c = Cc − αmN∗m +
1 − Fc(β∗
c)
fc(β∗
c)
; p∗m = Cm − αcN∗c +
1 − Fm(β∗
m)
fm(β∗
m)
. (4.5)
Note that D′i(x) = fi(x), i = c,m. This pricing rule shows that profit-maximizing fees
for each side of the market are equal to the cost of providing the service (Ci), adjusted
downward by the external benefit to the other side (αjN∗j ), and adjusted upward by a factor
((1 − Fi)/fi) that is related to the elasticity of participation.
To pin down the total number of transactions, the payment literature often assumes a
multiplicative relation, Nc,t · Nm,t, relying on an “independence” assumption. For the key
implications of our model, the choice of a specific functional form that describes the number
of transactions is not critical. What is essential, however, is the relatively mild assumption
that the value of virtual currency units necessary to make real transactions, i.e., TB∗t /V B∗
t ,
increases with the adoption of virtual currency by consumers and merchants.9
One specific form to implement this is the following. By relying on the widely used
multiplicative relation and assuming that the unit average value per transaction in terms
of the traditional currency is unaffected by the number of transactions, we obtain TB∗t =
T (Nc,t, Nm,t) = Nc,t · Nm,t. Additionally, we assume that the velocity of virtual currency is
proportional to the fraction of merchants who accept virtual currency, i.e., V B∗t = φ−1Nm,t,
for some constant φ > 0. In essence, this last assumption captures the notion that consumers
face more opportunities to spend their virtual currency units if more merchants accept virtual
currency. In other words, as the probability of a merchant accepting payments in virtual
currency increases, consumers may use their buffers of virtual currency more efficiently. This
effectively increases the velocity of virtual currency. Taken together, TB∗t = Nc,t · Nm,t and
V B∗t = φ−1Nm,t yield the total value of the units of virtual currency necessary to make real
9Formally, it is sufficient to assume TB∗t /V B∗
t = g(Nc,t, Nm,t), with either ∂g/∂Nc,t > 0 and ∂g/∂Nm,t ≥
0, or ∂g/∂Nc,t ≥ 0 and ∂g/∂Nm,t > 0. This generalization can be implemented by replacing φNc,t byg(Nc,t, Nm,t) and φN∗
c by g(N∗c , N
∗m) in subsequent equations.
15
payments, that is
TB∗t
V B∗t
= φNc,t. (4.6)
Using this expression in Eq. (3.5) gives the level of the exchange rate as an increasing
function of network adoption at any particular point in time as
Se /Bt =
φNc,t
MBt − ZB
t
. (4.7)
4.3 Speculative behaviour
We presume that the central bank is successful in the sense that changes in the price
level P e in terms of the traditional currency are anticipated with certainty. Speculators are
therefore assumed to choose their holdings of units of virtual currency while maximizing a
second-order approximation of their utility with respect to their next-period wealth, Wt+1,
where wealth is expressed in the established currency as
Us,t = E(Wt+1) −γ
2σ2(Wt+1), (4.8)
where E(Wt+1) and σ2(Wt+1) denote period’s t expectation and variance of wealth at time
t+ 1, respectively. For simplicity, we assume the same positive coefficient of risk aversion γ
across all speculators.10 The investment decision of the speculator is modelled as a trade-off
between investing in a risk-free bond denominated in the established currency and specu-
lation on virtual currency.11 Under the assumption of absence of lending and borrowing in
10This assumption can easily be relaxed by replacing the expression γ/Ns,t in the solution with the
expression
(
∑
i
1/γi
)−1
> 0, where γi 6= 0 denotes the coefficient of risk aversion of speculator i; see, e.g.,
Viaene and De Vries (1992). This modification does not change the essence of the model much. Bothexpressions measure, more or less, the “risk aversion aggregated across speculators.” We opt for the simplestexpression.
11We refer to Hirshleifer (1988) for a more sophisticated model of speculation with multiple investmentopportunities in a mean-variance framework. The results of Hirshleifer (1988) show that, given some fixedcosts for entering into a speculative position, the size of the speculative position depends on systematic riskand residual risk. In our study, the size of the speculative position will depend on the total risk.
16
virtual currency, the return on a position in virtual currency in terms of the established
currency is determined only by the change in the exchange rate. Hence, next period, the
wealth of a speculator investing in zBt units of virtual currency is
Wt+1 = R(Wt − Se /Bt zBt ) + S
e /Bt+1 zBt , (4.9)
where Wt is wealth at time t, R denotes the gross return on bonds denominated in the
traditional currency and where Se /Bt+1 denotes the uncertainty about the future exchange
rate Se /Bt+1 . We assume that individual speculators take the current exchange rate St as
given. In summary, the investor’s optimization problem is the maximization of (4.8) with
respect to zBt .
Solving the speculator’s optimization problem gives the optimal number of units of virtual
currency as
zBt =E(S
e /Bt+1 ) − RS
e /Bt
γσ2(Se /Bt+1 )
. (4.10)
The expression in Eq. (4.10) follows the standard solution in the literature for this type
of model, in which the expected additional return earned by a marginal increase in the
speculative position in the numerator is equal to the marginal decrease in utility owing to
additional risk taking. Summing the positions of Ns,t speculators in Eq. (4.10) gives the
aggregate speculative position in period t as
ZBt = Ns,tz
Bt =
E(Se /Bt+1 ) − RS
e /Bt
γNs,t
σ2(Se /Bt+1 )
. (4.11)
This equation gives the speculative demand for virtual currency as a function of the exchange
rate. Note that the aggregate speculative position cannot be negative (which would imply
17
money creation). Hence, Eq. (4.11) holds for E(Se /Bt+1 ) ≥ RS
e /Bt , or,
R−1E(S
e /Bt+1 ) ≥
TB∗t
MBt V B∗
t
.12 (4.12)
In other words, in the aggregate, speculators choose to take a positive position in virtual
currency only if the discounted expected value exceeds the hypothetical value of the current
exchange rate in the absence of speculation. Otherwise, they prefer to short sell the currency,
which they cannot do collectively, and their optimal aggregate position equals zero.
In summary, rewriting Eq. (4.11) gives the exchange rate at which speculators absorb
ZBt > 0 units of virtual currency as
Se /Bt =
(
E(Se /Bt+1 ) −
γ
Ns,t
ZBt σ
2(Se /Bt+1 )
)
R−1. (4.13)
The price that speculators are willing to pay for virtual currency equals the (discounted)
expected future exchange rate minus a risk premium for the uncertainty in the future value
of the speculative position in virtual currency.
4.3.1 Partial equilibrium
The level of the exchange rate is pinned down by Eqs. (3.5) and (4.13). These two
equations act, respectively, as supply and demand schedules for units of virtual currency.
Figure 3, panel (a) shows the two curves in a single diagram.
The upward-sloping curve of Eq. (3.5), derived from the price-quantity relation, shows
the price speculators have to pay to invest in an additional unit of virtual currency. The
larger the speculative position, the more units that are effectively withdrawn from circulation
in the payment system. Withdrawing units from circulation in the payment system results
in a price increase via the quantity relation. Hence, the larger the speculative position, the
higher the price to withdraw an additional unit from the virtual currency payment system.
12This is implied by ZBt ≥ 0 and Eq. (3.5).
18
Figure 3: Equilibrium exchange rate for virtual currency
(a) Low volume of real transactions (b) High volume of real transactions
0 4 8 12
04
812
MtBZt
B
E(S
t+1
€B) R
−1S
t€B
Speculative position Transactions
0 4 8 120
48
12Mt
BZtB
E(S
t+1
€B) R
−1S
t€B
Speculative position Transactions
The figure shows the demand of speculators for units of virtual currency in Eq. (4.11) and the available
units of virtual currency for speculators in Eq. (3.5) for different levels of the exchange rate, Se /Bt . The
equilibrium exchange rate follows from the intersection between the two lines and corresponds to the solutionin Eq. (4.13).
The downward-sloping curve of Eq. (4.13) shows the price that speculators would be
willing to pay, given their expectations, for a unit of virtual currency while investing in ZBt
units. The larger the speculative position ZBt , the more units speculators absorb, the greater
their risk, and the lower the price they are willing to pay to absorb an additional unit of
virtual currency.
The equilibrium exchange rate, Se /Bt , is determined by the intersection between the two
curves. It is only at this point on the upward-sloping curve that speculators have no incentive
to adjust the size of their positions.
If the condition in (4.12) is not satisfied, i.e., if current usage is sufficiently high, or
if speculators are sufficiently pessimistic, then the intersection coincides with the vertical
19
axis. The equilibrium exchange rate follows directly from Eq. (3.5) as speculators optimally
choose ZBt = 0.13
In contrast, if the condition in (4.12) holds because speculators are optimistic, or because
virtual currency usage is still low, then the analytical solution of the equilibrium exchange
rate is
Se /Bt =
1
2Se /B
t|TB∗
t =0+
√
(
1
2Se /B
t|TB∗
t =0
)2
+TB∗t
V B∗t
γ
Ns,tσ2(S
e /Bt+1 )R−1, (4.14)
where
Se /B
t|TB∗
t =0=
(
E(Se /Bt+1 ) −
γ
Ns,tMB
t σ2(Se /Bt+1 )
)
R−1. (4.15)
The equilibrium exchange rate in Eq. (4.14) is determined by two main components. The first
component, St|TB∗
t =0, represents the “purely speculative” exchange rate. It is the exchange
rate obtained from Eq. (4.13) when fixing ZBt = MB
t . This hypothetical exchange rate is
indicated in Figure 3, panel (a) by the intersection between the solid downward-sloping line
and the dashed vertical line, which specifies the total amount of available units of virtual
currency. We call it the purely speculative exchange rate, because it is the hypothetical price
speculators would be willing to pay for virtual currency if no real transactions are currently
settled using virtual currency, i.e., if TB∗t = 0.
The second component is the current value of virtual currency that is necessary to facil-
itate payments, i.e., TB∗t /V B∗
t . The larger the amount necessary for payments using virtual
currency, the higher the exchange rate. The more the value of virtual currency is absorbed
by payments, the lower the exchange rate risk that has to be absorbed by speculators, and
thus the higher the level of the exchange rate of virtual currency. Figure 3, panel (b) reports
the upward pressure on the current exchange rate from a greater number of payments using
13This is what would happen in the case of a traditional currency in a country facing extremely high andvolatile inflation. The assumption regarding perfect flexibility of prices with respect to the exchange ratemay be a reasonable characterization of such a case; see also the Introduction. The extremely inflationarynature eliminates the incentives of investors in the model to hold such a currency as a store of value becauseof the low value of the expected exchange rate.
20
virtual currency by showing the new equilibrium after an increase in TB∗t /V B∗
t , causing a
shift in the upward-sloping curve.
4.3.2 Impact of speculative environment
Virtual currency has suffered from highly volatile exchange rates compared with the
exchange rates of established currencies; see, e.g., Yermack (2015). The high level of volatility
is often attributed to the behaviour of speculators. Speculators herding towards a new
opportunity and erratic changes in their beliefs may cause large swings in the exchange rate.
Of course, the mere presence of speculators cannot explain the higher level of volatility, since
speculators may invest in any currency. Nevertheless, it is possible that the exchange rate is
especially sensitive to changes in speculators’ beliefs in the early-adoption phase, when few
real payments are settled in virtual currency.
The impact of speculators’ beliefs regarding the exchange rate can be assessed by taking
the derivative of Eq. (4.14) with respect to the speculators’ expectations of the future
exchange rate as
∂Se /Bt
∂E(Se /Bt+1 )
=R−1
2+
R−1
2
1
2Se /B
t|TB∗
t =0√
(
1
2Se /B
t|TB∗
t =0
)2
+TB∗
t
V B∗
t
γNs,t
σ2(Se /Bt+1 )R−1
≤1
R. (4.16)
This equation shows which determinants affect the change in the exchange rate following a
shock to speculators’ beliefs about the future value of virtual currency.
Shocks in the expectations of speculators as to the future value have a larger impact
on the exchange rate in the early phase of a virtual currency. From Eq. (4.16), it is not
difficult to derive that ∂Se /Bt /∂E(S
e /Bt+1 ) = R−1 if TB∗
t = 0. Hence, in the absence of any
real transactions, changes in the beliefs of speculators on the future exchange rate translate
one-to-one into discounted changes in the current exchange rate. The impact of changes
in the beliefs of speculators is strictly decreasing in the value of virtual currency that is
currently necessary to facilitate payments TB∗t /V B∗
t . Hence, the impact of actions based on
21
changes in speculators’ beliefs regarding the future value of virtual currency is expected to
become smaller once a virtual currency is more widely used. This can also be observed from
Figure 3. An improvement in speculators’ beliefs about the future value of the exchange
rate is represented by an upward shift of the downward-sloping curve. The resulting change
in the equilibrium exchange rate is larger in the low-transaction-volume environment, panel
(a), than in the high-transaction-volume environment, panel (b).
The larger impact of speculators’ beliefs about the exchange rate in the early phase of a
virtual currency can be intuitively explained as follows. In the absence of real transactions,
any adjustment in the exchange rate to the new expectations is completely in the price
domain. In the presence of real transactions, however, the adjustment towards the new
equilibrium will partly be in the quantity domain as well. A more pessimistic view about
the exchange rate by speculators will reduce the exchange rate. Given the value of real
transactions, such a reduction in the exchange rate will simultaneously require an increase
in the number of the units of virtual currency used to facilitate real transactions. This
implies a lower number of units of virtual currency in the hands of speculators in the new
equilibrium. The larger the value of real transactions using virtual currency, the larger the
quantity effect and, hence, the smaller the price change.
In a similar way, it can be shown that the entry of speculators drives up the current
exchange rate. In the model, this is equivalent to the entry of new speculators or having
current speculators take on more risk: both correspond to a decrease in risk aversion aggre-
gated across speculators, i.e., γ/Ns,t. Given the expectations, the entry of new speculators
turns the downward-sloping line in Figure 3 counter-clockwise at the intersection with the
y-axis. Taking the derivative of Eq. (4.14) with respect to the speculators’ aggregated risk
aversion yields
∂Se /Bt
∂ γNs,t
=(
MBt S
e /Bt − TB∗
t /V B∗t
) −σ2(Se /Bt+1 )R−1
2
√
(
1
2Se /B
t|TB∗
t =0
)2
+TB∗
t
V B∗
t
γNs,t
σ2(Se /Bt+1 )R−1
. (4.17)
22
This expression is negative for any γ/Ns,t: The first term in parentheses is positive, because
we have MBt ≥ TB∗
t /(V B∗t S
e /Bt ) from Eq. (3.5). In other words, an inflow of more speculators
or an increased risk appetite are both expected to increase the exchange rate of virtual
currency.
The inflow and outflow of speculators in the virtual currency market have a larger impact
on the exchange rate in the early-adoption phase of virtual currency. Although the derivative
in (4.17) remains negative, it gets closer to zero as the transaction volume TB∗t /V B∗
t increases.
In other words, the model suggests that the exchange rate becomes less sensitive to the
inflow and outflow of speculators as the use of virtual currency by merchants and consumers
increases in intensity. To summarize, the impact of speculative behaviour on the exchange
rate of virtual currency becomes smaller as virtual currency is used more frequently to make
real payments.
5 Rational Equilibrium Exchange Rate Analysis
The (partial) equilibrium for the exchange rate derived in Eq. (4.14) does not rely on an
assumption of rational expectations on the part of speculators. In this section, we show how
the rational equilibrium exchange rate can be obtained by combining this specification with
the payment network equilibrium outcome.
When the number of transactions has reached its stationary equilibrium (and no more
growth can be expected), the condition in (4.12) implies that speculators will no longer
have an incentive to invest in virtual currency. Hence, all available units will be used to
purchase real goods and services, i.e., ZBt+1 = 0. While assuming rational expectations
among speculators, Eq. (4.7) implies that the expected future exchange rate in period t
equals
E(Se /Bt+1 ) = q
(
φN∗c
MBt+1
)
, (5.1)
where MBt+1 = (1 + mB
t+1)MBt is the number of virtual currency units at t + 1 that follows
23
from the predetermined growth rule. Moreover, the volatility of the future exchange rate
equals
σ2(Se /Bt+1 ) = q(1 − q)
(
φN∗c
MBt+1
)2
. (5.2)
In other words, the larger the success probability q, the larger the expected exchange rate,
and, conditional upon q > 1/2, the lower the volatility of the future exchange rate. Sub-
stitution of Eqs. (4.7), (5.1) and (5.2) into the partial equilibrium for the current exchange
rate in Eq. (4.14) gives
Se /Bt = q
(
φN∗c
MBt+1
)
×
(
1
2
√
δ2t +1
2
√
δ2t + 4γφNc,t
Ns,t
1 − q
qR−1
)
. (5.3)
The first term in Eq. (5.3) is the expected future exchange rate. Its value depends on the
success probability q, and on the – conditional upon success – expected long-term growth in
the adoption of virtual currency towards N∗c .
The second term in Eq. (5.3) is the “discount factor” applied by investors. The δt
represents the hypothetical discount factor in case no real transactions are settled using
virtual currency, i.e., if Nc,t = 0. This hypothetical discount factor is calculated as
δt =
(
1 −(1 − q)
1 + mBt+1
γφN∗
c
Ns,t
)
R−1. (5.4)
The full equilibrium equation in (5.3) shows that the actual discount factor is an increas-
ing function of current adoption of virtual currency Nc,t.14 Actual adoption increases the
exchange rate of virtual currency via this channel. Moreover, the exchange rate of virtual
currency does not suffer from “money illusion.” Given all other parameters in Eqs. (5.3)
14It is also possible to derive a, albeit somewhat less elegant, closed-form solution for the exchange rateunder a (CRRA) specification assuming log-utility for speculators. Such an alternative specification leavesour results qualitatively unchanged, but requires a mild condition of having sufficient current (aggregate)wealth Wt in the hands of speculators relative to the current value of virtual currency necessary to facilitatepayments TB∗
t /V B∗t .
24
and (5.4), doubling the number of virtual currency units MBt+1 reduces the exchange rate
Se /Bt by one-half.
6 Discussion
In this section, we discuss several extensions of the model.
6.1 Speculating consumers
Speculators and users of virtual currency making actual payments are assumed to be
different agents in the model. In practice, speculators and users are sometimes the same
agents; see, e.g., Johnson (1960). This does not change the essence of the model. We show
this by considering the case of speculation on the consumer side. Suppose that the decision
to join the network provides consumers with the opportunity to speculate on the value of
virtual currency. Presuming that users face the same investment decision as speculators,
then their net utility derived from using the monopolistic virtual currency payment network
would be
Uc,t = αcNm,t + βc + ∆c,t − pc,t, (6.1)
where ∆c,t is the additional utility derived from the opportunity to speculate on the value
of virtual currency, which equals
∆c,t = zBt E(Se /Bt+1 − St) −
γ
2σ2(zBt S
e /Bt+1 ), (6.2)
for the optimal level of the speculative position zBt , which is derived in Eq. (4.10).15
The steady-state equilibrium will not be changed by giving consumers the opportunity to
speculate on the value of virtual currency. The reason is that the additional utility derived
15Basically, the extension here is that consumers are no longer constrained to zBt = 0 in their investmentdecision. The value of ∆c,t in Eq. (6.2) is derived from the difference between Eq. (4.8) for any zBt > 0 andfor zBt = 0. Differences in the degree of risk aversion, e.g., between speculators and consumers, can be dealtwith as described in footnote 10.
25
from the opportunity to speculate on the value of virtual currency ∆∗c = 0 equals zero in
the steady state, since no further appreciation of virtual currency is to be expected. In the
steady state, both speculators and consumers will reduce their positions held for speculation
to zero, and therefore the expression in (6.2) will equal zero. If the utilities of the users are
unchanged, the equilibrium number of users, N∗c and N∗
m, will also be unchanged. Therefore,
the future exchange rate will be unchanged.
Nevertheless, the opportunity for consumers to speculate on the value of virtual currency
will affect the current exchange rate. This is because the aggregate speculative position of
both speculators and consumers will be equal to zBt (Ns,t +Nc,t). When working through the
entire equilibrium solution, this basically means that the Ns,t in the denominators in Eqs.
(5.3) and (5.4) will be replaced by Ns,t + Nc,t.16 In other words, the additional speculative
demand from the consumer side will increase the discount factor and, therefore, provide
upward pressure on the level of the current exchange rate.
The opportunity of consumers to speculate may change the dynamics of the model. In
terms of comparative statics for the current exchange rate, an increase in the consumer base
of the virtual currency network will not only increase the value of the transactions using
virtual currency, but will also increase speculative demand. In other words, both curves in
the diagrams in Figure 3 will shift upward as a consequence of more consumers using virtual
currency. Such an increase in consumer usage is illustrated in Figure 4. Besides the upward
shift of the upward-sloping curve as a result of higher transactional demand stemming from
more consumers using virtual currency, the inflow of new speculating consumers also turns
the downward-sloping schedule representing speculative demand counter-clockwise. This
further increases the current exchange rate. The counter-clockwise turn of the speculative
demand schedule shown in Figure 4 does not presume higher expectations regarding the
exchange rate as new consumers enter the virtual currency network. Such a change in
16Here, Ns,t = 0 corresponds to the special case in which all speculators on the value of virtual currencyalso use the virtual currency to make payments.
26
Figure 4: Equilibrium exchange rate with speculating consumers
0 4 8 12
04
812
MtBZt,old
B
E(S
t+1
€B) R
−1S
t,old
€B
Zt,newB
St,n
ew€
B
The figure shows the shift in the upward-sloping curve representing transactional demand and the shift inthe downward-sloping speculative demand schedule as a result of more consumers adopting virtual currencyif consumers are allowed to take a speculative position in virtual currency.
expectations would shift the speculative demand schedule further upward, and would result
in a reinforced increase in the current exchange rate.
Moreover, allowing speculating consumers results in a higher level of utility derived from
joining the network for early adopters, since these consumers do not only benefit from using
virtual currency, but also from a non-zero speculative position in virtual currency. Therefore,
higher expectations regarding the future value of virtual currency will not only increase the
positions of speculators, but also draw more consumers towards the network, which may
result in a faster convergence to the steady-state use of virtual currency.
6.2 Intra-group benefits
Positive intra-group externalities due to, e.g., peer-to-peer payments, such as remittances,
do not change the essence of the model. Although consumers are not directly affected by
other consumers in the original specification, in equilibrium, consumers exert a positive (in-
27
direct) externality on each other through their impact on merchant acceptance. To explicitly
account for positive intra-group externalities, we may modify the net utility from joining the
virtual currency network to
Ui,t = αiNj,t + αiiNi,t + βi − pi,t, i, j = c,m, i 6= j. (6.3)
Here, the parameter αii > 0 denotes the benefit that user i enjoys from interacting with other
(same type) users on the network. Essentially, given prices pc,t and pm,t, the extension in Eq.
(6.3) yields two demand equations, Nc,t = Nc(pc,t, Nm,t, Nc,t) and Nm,t = Nm(pm,t, Nc,t, Nm,t),
in two unknowns (Nc,t, Nm,t). Solving this system gives N∗c,t(pc,t, pm,t) and N∗
m,t(pc,t, pm,t).
Substituting these optimal demand functions into the network’s profit function in Eq. (4.3)
and maximizing then yields optimal fees p∗c,t and p∗m,t. Qualitatively, this extension will not
alter our results.
7 Concluding Remarks
This paper proposes an economic framework for analyzing the exchange rate of a virtual
currency and its key drivers. The paper documents three main determinants. First, the cur-
rent use of virtual currency to make real payments. Second, the decision of forward-looking
investors to buy virtual currency (thereby effectively reducing its supply) given expected
growth in virtual currency transactions. Third, the elements that jointly drive consumer
adoption and merchant acceptance of virtual currency, which determine expected long-term
growth in usage. In particular, we show that the equilibrium exchange rate depends on
both a purely speculative component pinning down a “floor” under the exchange rate and
a transaction component that affects the exchange rate risk absorbed by speculators. More
widespread use of virtual currencies by merchants and consumers lowers the impact of spec-
ulative behaviour and therefore stabilizes the exchange rate.
Our analysis illustrates that a steep increase in the exchange rate due to speculative
28
motives is exactly what one can expect at the introduction of a potentially successful virtual
currency. Moreover, the current high levels of volatility seem to be a symptom of early
development: theoretically, volatility is expected to drop if the adoption by consumers and
merchants increases. The future will show how much volatility will drop. The fixed supply
of virtual currency may suggest its volatility will reflect that of other commodities (see, e.g.,
Selgin, 2014) rather than that of traditional currencies whose quantities are managed by
central banks. Moreover, the model also shows that, conditional upon survival, deflationary
virtual currency prices may be expected during the early-adoption stage.
Our study is only a first step in trying to understand the underlying economics of virtual
currencies. Empirically, little is known about the actual number of payments in virtual cur-
rency for goods and services. Reliable time series for user and acceptance statistics are still
lacking and need to be developed, as these will be – without a doubt – an important input for
studying the behavioural aspects of using virtual currencies. Competition among virtual cur-
rencies raises a further issue of whether only a few virtual currencies will ultimately dominate
a global market where network effects play an important role. Our framework suggests that
the exchange rates of various virtual currencies may diverge widely, depending on adoption,
transactional demand, the quantity and growth rate of monetary units, speculative demand
and network stability (survival probabilities). Finally, analyzing the empirical determinants
of switching behaviour from traditional means of payment to virtual currencies may lead to
interesting cross-fertilizations between the payment choice literature and the literature on
currency substitution. Much more research remains to be done, and we are planning to go
down that route.
29
Appendix A: Estimate of the Speculative Position
The hypothetical exchange rate in the absence of speculation in Figure 1 is based on a very
rough estimate of the magnitude of the speculative position. Theoretically, the magnitude of
speculative position is defined as the number of units of virtual currency not used to settle
payments for goods and services. A rough estimate of this amount is obtained as follows. At
each day t, we obtain from the public ledger the number of bitcoins associated with addresses
that are not involved in any transactions over the next three months. This is the empirical
proxy for Zt in Eq. (3.5). Given this estimate of the speculative position on day t, the total
number of bitcoins in circulation on day t, and the actual exchange rate on day t, it is not
difficult to calculate from Eq. (3.5) the hypothetical exchange rate for a speculative position
with a zero balance, while assuming a fixed ratio TB∗t /V B∗
t . Figure 1 presents this level as
the grey line. The dashed line is based on an exponentially weighted average with a daily
smoothing parameter of 0.997 (this corresponds to an observation one quarter ago receiving
a 75 percent weight of the weight of the most recent observation).
We described our estimate of the size of the speculative position as a rough estimate. The
reason is that there are several issues with this estimate. Our approach relies on subtracting
the bitcoins not involved in transactions for real goods and services from the total amount.
Even though all transactions are recorded in the public ledger, this does not directly reveal
which units of virtual currency have not been used to settle payments for real goods and
services. All recorded transactions represent a larger set than the transactions for processing
payments for real goods and services, since they may include, among others, transactions
with exchanges, transactions with other speculators, and transactions between addresses
without a change in ownership (i.e., someone moving bitcoins from the left to the right
pocket). In this respect, our approach is likely to result in an overestimation of the number
of bitcoins involved in processing payments for real goods and services, and, therefore, in an
underestimation of the size of the speculative position. In this context, the level of the grey
line in Figure 1 is probably too high.
30
Moreover, the raw data on the total number of bitcoins in circulation is not corrected
downward for units that are effectively permanently withdrawn from circulation. This refers
to units that are permanently lost because users lost access to the cryptographical keys that
are necessary to transfer bitcoins from one address to another. Of course, the number of
Bitcoin units that are truly permanently lost is difficult to measure. However, if such data
were available, following Eq. (3.5), a correction based on such data would result in an upward
adjustment of the exchange rate in the absence of speculation.
31
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